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The INdAM special period in Padova consists of two intensive months devoted to the study of the Langlands Program.  The activities of this period will consist of regular seminars and minicourses by visitors, and a school

The Langlands program is a centerpiece of research in modern number theory incorporating and animating research in such diverse fields as representation theory, complex analysis, arithmetic geometry, and algebraic topology. Its ultimate goal is to relate analytic objects (automorphic forms) with objects of algebraic nature (Galois representations). Progress in this area have been extremely fast in the past 20 years and have received the highest interest by the international scientic community, see for instance the talks at the ICM given by L. Lafforgue (2002), N. Chau, C. Breuil (2010), M. Emerton, J. Arthur, M. Harris (2014).

The school will take place is late Spring 2019, aimed at introducing a large audience of Ph.D students and young researchers to a recent and exciting development of this program -the so called p-adic local Langlands, pioneered by C. Breuil, P. Colmez and M. Emerton since the early 2000s. The p-adic local Langlands should be read as a tremendous far-reaching generalization of the insight of the Tanyiama-Weil conjectures and Serre's modularity conjectures and its establishment for the particular case of GL2(Qp) led to spectacular applications such as the proof of the Fontaine-Mazur conjecture. The school will be organized in a series of lectures, starting from a Ph.D level and given by some of the most established experts in this area, encouraging the interaction between the lecturers and the students and aiming at initiating new collaborations and research projects.